Discrete optical soliton scattering by local inhomogeneities

Lasha Tkeshelashvili a,b,*a Andronikashvili Institute of Physics, Tamarashvili 6, 0177 Tbilisi, Georgia

b Tbilisi State University, Chavchavadze 3, 0128 Tbilisi, Georgia

Received 27 July 2012; received in revised form 10 October 2012; accepted 30 October 2012

Available online 9 November 2012

Abstract

The nonlinear wave scattering by local inhomogeneities in discrete optical systems is studied both analytically and numerically.

The presented theory describes the reflection and transmission of discrete optical solitons at a point defect. In particular, the derived

expressions determine the reflected and transmitted pulses from the incident one. In the range of validity, the analytical results are in

excellent agreement with the numerical simulations. It is demonstrated that the point defects in structured optical materials

represent effective tool for controlling and manipulation of the nonlinear light pulses.

# 2012 Elsevier B.V. All rights reserved.

Keywords: Discrete optical solitons; Point defects; Wave scattering

www.elsevier.com/locate/photonics

Available online at www.sciencedirect.com

Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101

1. Introduction

The studies of discrete wave dynamics in structured

systems range from the realization of optical analogies of

various quantum-mechanical phenomena [1,2] to prac-

tical design of functional elements for the all-optical

communication networks [3,4]. In particular, the tailored

light-matter interaction processes in such systems [5,6]

allow to demonstrate unique effects related to the

nonlinear optical pulses called solitons [7].

Solitons are localized wave packets that can

propagate undistorted in homogeneous nonlinear

media. That peculiarity makes nonlinear systems very

attractive for applications in the field of all-optical

communications [8]. However, in general, the soliton

interaction with inhomogeneities is an extremely com-

* Correspondence address: Andronikashvili Institute of Physics,

Tamarashvili 6, 0177 Tbilisi, Georgia. Tel.: þ995 32 239 87 83;

fax: þ995 32 239 14 94.

E-mail address: lashat@mail.com.

1569-4410/$ – see front matter # 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.photonics.2012.10.001

plicated process. Indeed, the solitons represent solutions

of so-called integrable nonlinear equations [9]. The

inhomogeneity, at least locally, breaks the integrability of

the model. In the non-integrable models the stability of

nonlinear pulses is not guaranteed anymore, and the non-

elastic effects such as the soliton radiative decay in the

scattering processes may take place [10].

Perhaps, the effectively one-dimensional discrete

structures represent the most convenient systems for

study of the nonlinear wave dynamics [1,7,11]. In

particular, much of the important theoretical and

experimental results were obtained for arrays of optical

waveguides [12], coupled nano-cavities in photonic

crystals [13], metallo-dielectric systems [14,15], and the

Bose–Einstein condensates in deep optical lattices [16–

18]. The universal mathematical model that governs the

wave dynamics in such systems is the Discrete Nonlinear

Schrodinger (DNLS) equation [19,20]:

i@cn

@tþ Cðcnþ1 þ cn�1Þ þ Njcnj2cn þ encn ¼ 0:

(1)

L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–10196

Depending on the system under consideration, t is either

the temporal or spatial variable [1]. In the case when t is

the temporal variable, the localized wave packets are

the optical pulses. However, for the systems such as the

coupled waveguide arrays, t is a spatial coordinate, and

those pulses represent optical beams. The eigenmode

amplitude at the site n is cn, and C gives the evanescent

coupling rate between adjacent sites. N is the nonlinear

coefficient. Here, the inhomogeneity is introduced

through the n-dependent en. In different cases that might

reflect different physical factors. For instance, in the

case of arrays of optical waveguides, the variation of en

from site to site may be caused by the different refrac-

tive index of the individual waveguides.

It should be noted that the defect states can be

introduced in the system by different means. In

particular, the bond defects defined as a local variation

in C were studied in [21–24]. Moreover, Ref. [25]

addressed the effects caused by local inhomogeneities

in the nonlinear coefficient N. The problem of soliton

scattering by localized defects in binary optical lattices

was considered in [26]. Below, following [27,28], it is

assumed that C, as well as the nonlinear coefficient N, is

constant for all sites. Thus, only the linear term

associated with en defines a scatterer in the system.

Nevertheless, even in such case the nonlinear effects

may cause coupling to the linear defect modes [29].

Indeed, the numerical simulations reported in [27,28]

show that various scattering regimes might be realized

with soliton trapping, reflection, and transmission

effects. In what follows, the weakly nonlinear wave

scattering by point defects in the discrete optical

systems is studied both analytically and numerically. In

this case, the nonlinear term in Eq. (1) can be treated as

a small perturbation to the linear part. That allows to

derive analytical expressions for the scattered waves.

Note that, only the variation in en is relevant for the

wave propagation and scattering processes. Indeed, by

means of cn! expðietÞcn transformation, the values of

en in the corresponding term of Eq. (1) can be shifted by

an arbitrary constant e. Suppose that the point defect is

located at n = 0. Then, without loss of generality, en for

n 6¼ 0 can be assumed to be exactly zero, i.e. en = e d0,n.

Here, d0,n is the Kronecker delta, and e determines the

defect strength.

2. Perturbation analysis

In the weakly nonlinear limit the problem can be

simplified significantly by means of the reductive

perturbation method [30,31]. In particular, the non-

integrable Eq. (1) can be reduced to the integrable

Nonlinear Schrodinger (NLS) equation for the carrier

wave envelope [7,8]. In turn, the inverse scattering

method allows to obtain the solutions of the NLS model

analytically [9]. In particular, under certain conditions,

the NLS equation supports the bright soliton solutions.

Physically, the soliton formation is possible when the

balance between the nonlinear effects and the linear

dispersion (or diffraction) takes place. That happens

when the wave envelope is a slowly varying function on

the carrier wavelength scale. It must be stressed that the

wider envelope soliton has the smaller amplitude [8,9].

Therefore, the presented theory describes the dynamics

of solitons which are sufficiently broad.

As a result of the localized pulse scattering process, in

the final state, there exist the reflected wave propagating

backwards, and the transmitted wavewhich tunnels to the

other side of the defect. In general, the incident pulse can

excite the localized defect mode as well [1,7,32]. That

may cause the trapping of the incident soliton by the

defect [27,28]. However, as will be shown below, such

bound states are not realized for the weakly nonlinear

pulses and, therefore, can be neglected. Moreover, since

the point defect is assumed to be linear, in the reflection

process the nonlinear frequency conversion processes do

not take place. Therefore, the incident, reflected, and

transmitted waves have the same wave numbers |k|.

2.1. Transmitted wave

Suppose that the incident localized pulse impinges at

the point defect from the left region (n < 0). Then,

according to Ref. [31], the ansatz for the transmitted

wave in the right region n > 0 is:

cn ¼X1m¼1

Xl�m

mmV ðm;lÞðj; tÞEðlÞn : (2)

Here, the smallness parameter m � 1 guarantees that

the nonlinear term in Eq. (1) can be treated as a small

perturbation. Moreover, in order to describe explicitly

the wave envelope dynamics, the set of new ‘‘slow

variables’’, t = m2n and j ¼ mðn � vgtÞ, is introduced.

The carrier wave is EðlÞn ¼ expðil½kn � vt�Þ, and v is

given by the dispersion relation:

v ¼ �2C cosðkÞ: (3)

In addition, vg is the group velocity:

vg ¼dv

dk¼ 2C sinðkÞ: (4)

The perturbation analysis based on Eqs. (1) and (2)

results in the NLS equation for the slowly varying

L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101 97

envelope V(1,1)(j, t) [30,31]:

iV ð1;1Þt þ Dk

2Vð1;1Þjj þ NkjVð1;1Þj2Vð1;1Þ ¼ 0; (5)

with

Dk ¼ cotðkÞ; (6)

and the nonlinear coefficient reads:

Nk ¼1

2

N

C sinðkÞ : (7)

The slow variables in the subscripts represent the

corresponding partial derivatives, i.e, ft � @f/@t, etc.

As can be seen from the definition of j, Eq. (5)

describes the transmitted wave in the reference frame

moving with the group velocity vg.

2.2. Reflected wave

In the left region there is only the incident pulse

initially. However, the scattering process results in the

reflected wave as well. Therefore, for n < 0 the ansatz

for cn reads [31]:

cn ¼X1m¼1

Xlþl0�m

mmUðm;l;l0Þðj; j; tÞEðl;l0Þn ; (8)

here Eðl;l0Þn ¼ expðil½kn � vt� � il0½kn þ vt�Þ. The addi-

tional slow variable j ¼ mðn þ vgtÞ is introduced to

describe the reflected wave envelope. Nevertheless, at

this stage of calculations, Uðm;l;l0Þ envelopes are allowed

to be functions of all (that is t, j and j) variables.

Inserting Eq. (8) into Eq. (1) reduces the initial

DNLS equation the NLS model [30,31]. In particular,

the perturbation analysis shows that the slowly varying

envelope of incident wave does not depend on j (i.e.

U(1,1,0) = U(1,1,0)(j, t)) and:

iUð1;1;0Þt þ Dk

2Uð1;1;0Þjj þ NkjUð1;1;0Þj2Uð1;1;0Þ ¼ 0: (9)

Furthermore, the envelope of the reflected wave is inde-

pendent of j (i.e. Uð1;0;1Þ ¼ Uð1;0;1Þðj; tÞ) and obeys:

iUð1;0;1Þt � Dk

2Uð1;0;1Þjj

� NkjUð1;0;1Þj2Uð1;0;1Þ ¼ 0: (10)

Eqs. (9) and (10) are written in the reference frames

moving with the group velocities vg and �vg, respectively.

2.3. Continuity conditions

The above analysis shows that the nonlinear wave

dynamics on both sides of the point defect is governed

by the NLS equation. The equation of motion at n = 0

can be used relate the amplitudes of the transmitted and

reflected waves with that of the incident wave. Indeed,

since the nonlinear term is small, at the defect Eq. (1)

approximately gives:

vc0 þ Cðc1 þ c�1Þ þ ec0 � 0: (11)

Here, c0 is given by Eq. (2):

c0 �X1m¼1

Xl�m

mmV ðm;lÞEðlÞ0 ; (12)

However, in addition, according to Eq. (8):

c0 �X1m¼1

Xlþl0�m

mmUðm;l;l0ÞEðl;l0Þ0 : (13)

Eq. (12) in combination with Eq. (13) results in:

V ð1;1Þ ¼ Uð1;1;0Þ þ Uð1;0;1Þ: (14)

Note that at n = 0 the slow variables are:

t ¼ 0;j ¼ �mvgt;j ¼ þmvgt:

Furthermore, Eq. (8) for c�1 gives:

c�1 ¼X1m¼1

Xlþl0�m

mmUðm;l;l0ÞEðl;l0Þ�1 ; (15)

and Eq. (2) yields the following expression for c1:

c1 ¼X1m¼1

Xl�m

mmV ðm;lÞEðlÞ1 : (16)

At n = �1 the slow variables read:

t ¼ �m2;j ¼ mð�1 � vgtÞ;j ¼ mð�1 þ vgtÞ:

Furthermore, the slowly varying functions V(1,1),

U(1,1,0), and U(1,0,1) are expanded in the Taylor series:

V ð1;1Þð�mvgt � m; �m2Þ � Vð1;1Þð�mvgt; 0Þ;Uð1;1;0Þð�mvgt � m; �m2Þ � Uð1;1;0Þð�mvgt; 0Þ;Uð1;0;1Þðþmvgt � m; �m2Þ � Uð1;0;1Þðþmvgt; 0Þ:

Since m � 1, in these expansions only the leading terms

are retained.

The expressions for the reflected and transmitted

wave envelopes at n = 0 follow from Eqs. (11) and (14):

Uð1;0;1Þðj; 0Þ ¼ R Uð1;1;0Þð�j; 0Þ; (17)

L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–10198

Fig. 1. The soliton scattering process at the local inhomogeneity.

Here, |cn(t)| is plotted. In this simulation the point defect is located at

n = 2000 (shown by the dotted line). The values of other parameters

and further details are given in the text.

with the reflection coefficient R:

R ¼ � ee þ 2iC sinðkÞ ; (18)

and

Vð1;1Þðj; 0Þ ¼ T Uð1;1;0Þðj; 0Þ; (19)

with the transmission coefficient T:

T ¼ 2iC sinðkÞe þ 2iC sinðkÞ : (20)

Eqs. (17) and (19) determine the reflected and transmit-

ted waves from the incident pulse. Moreover, it is easy

to see that |R|2 + |T|2 = 1. That is in full agreement with

the energy conservation law since the wave trapping

processes by the defect are neglected.

2.4. Bound states

It should be noted that �p � k � p [1]. Never-

theless, according to Eqs. (17) and (19), R and T diverge

for the imaginary value k = ik. Here, k obeys:

sinhðkÞ ¼ e2C

: (21)

As with the quantum mechanics scattering problems,

the divergence of the reflection and transmission coef-

ficients is due to the discrete energy level associated

with the point defect. That level gives rise to the bound

state. The frequency v = V, which corresponds to k,

follows from Eqs. (3) and (21):

V ¼ �2C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ e

2C

� �2r

: (22)

The frequency range of the modes with real k does not

overlap with V. Nevertheless, nonlinear effects may

cause the strong localization of the pulse in real space.

In this case, the Fourier spectrum of the wave packet

may become broad enough to be in resonance with V.

The weakly nonlinear pulses are not strongly localized,

and therefore, the processes which lead to the bound

state formation do not take place. In this context, it must

be stressed that the group velocity vg is an important

parameter as well [27].

3. Results

Let us consider the soliton scattering by the point

defect in more details. A numerical example of such

process is shown in Fig. 1. As it is demonstrated above,

the dynamics of the weakly nonlinear pulses is governed

by the celebrated NLS equation:

i@F

@zþ P

2

@2F

@x2þ QjFj2F ¼ 0: (23)

The solutions of the NLS model can be obtained by

means of the inverse scattering method [8,9]. In partic-

ular, Eq. (23) supports the bright soliton solutions if

PQ > 0. Therefore, since it is assumed that the bright

solitons can propagate in the system, the values of k are

further restricted by DkNk > 0 inequality. For instance,

if C and N are simultaneously positive, |k| < p/2 as can

be seen from Eqs. (6) and (7). The presented theory

describes p/2 < |k| < p range too. However, for those

values of k the stable optical pulses do not exist, and

therefore, such cases will not be considered below.

3.1. Fundamental soliton scattering

Suppose that the incident pulse represents the

fundamental soliton solution of Eq. (9):

Fðj; tÞ ¼ 1

L

ffiffiffiffiffiffiDk

Nk

rsech

j

L

� �exp i

Dk

2

t

L2

� �: (24)

In this expression a real parameter L is the soliton width

[9]. However, L is related to the soliton amplitude as well.

In particular, to wider solitons correspond the smaller

values of the amplitude. As was discussed above, for the

given vg, the value of L must be sufficiently large to

suppress the formation of the bound state.

From Eqs. (17) and (19) it directly follows that the

initial conditions for Eqs. (10) and (5) read:

FRðj; 0Þ ¼ R

L

ffiffiffiffiffiffiDk

Nk

rsech

j

L

� �; (25)

– Fundamentals and Applications 11 (2013) 95–101 99

FTðh; 0Þ ¼ T

L

ffiffiffiffiffiffiDk

Nk

rsech

j

L

� �; (26)

Eqs. (25) and (26) define the initial value problem for

the reflected and transmitted waves, respectively.

The sech-type initial value problem of the NLS

equation was solved in Ref. [33]. In particular, for the

following initial condition:

Fiðx; z ¼ 0Þ ¼ A

L

ffiffiffiffiP

Q

rsech

x

L

� �; (27)

the number of generated solitons in Eq. (23) is the

maximum integer M which satisfies:

M < jAj þ 1

2: (28)

For |A| � 1/2, no soliton emerges and the pulse disperses.

The dispersive modes, similar to the linear waves, have

vanishing amplitudes in the final state. If 1/2 � |A| � 3/2,

Eq. (28) shows that M = 1. In this case, by sheding the

dispersive radiation, the solution asymptotically relaxes

to the fundamental soliton with the amplitude:

AðsÞ ¼ 2

L

ffiffiffiffiP

Q

rjAj � 1

2

� �: (29)

In general, M > 1 and there emerges the bound state of

M solitons plus the dispersive radiation [8,33]. Howev-

er, such higher-order soliton solutions are not relevant

for the present problem.

It can be seen from Eqs. (25) and (27) that for the

reflected wave A = AR, where:

AR ¼ R: (30)

Similarly, from Eqs. (26) and (27), for the transmitted

wave follows A = AT:

AT ¼ T : (31)

Furthermore, Eqs. (18) and (20) give:

jARj ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ s�1p ; (32)

jAT j ¼ 1ffiffiffiffiffiffiffiffiffiffiffi1 þ sp ; (33)

where s represents the effective strength of the defect:

s ¼ e2C sinðkÞ

� �2

: (34)

Note that the lower bound for s is smin = e2/(2C)2. Eq. (34)

gives the quantitative measure of the group velocity vg (see

Eq. (4)) influence on the scattering process.

L. Tkeshelashvili / Photonics and Nanostructures

According to Eqs. (28) and (32) there is no soliton in

the reflected pulse for 0 < s < 1/3. Moreover, for

s > 1/3, only one reflected soliton is generated with the

amplitude:

AðsÞR ¼

1

L

ffiffiffiffiffiffiDk

Nk

r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ s�1p � 1

� �: (35)

By comparing with Eq. (24), the width of the reflected

soliton LR reads:

LR ¼ L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ s�1p

2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ s�1p : (36)

For the transmitted wave Eqs. (28) and (33) show

that there exists only dispersive radiation for s > 3. If

0 < s < 3 one soliton emerges in the transmitted pulse

with the amplitude:

AðsÞT ¼

1

L

ffiffiffiffiffiffiDk

Nk

r2ffiffiffiffiffiffiffiffiffiffiffi

1 þ sp � 1

� �; (37)

and comparison with Eq. (24) shows that the width of

the transmitted soliton LT is:

LT ¼ L

ffiffiffiffiffiffiffiffiffiffiffi1 þ sp

2 �ffiffiffiffiffiffiffiffiffiffiffi1 þ sp : (38)

These expressions show how the incident soliton am-

plitude and width transform in the scattering process.

For example, for s = 1, the reflected and the transmitted

solitons are approximately 2.414 times wider compared

to the incident pulse.

3.2. Numerical simulations

An example of the fundamental soliton scattering

process at the point defect is depicted in Fig. 1. The

presented result is the numerical solution of Eq. (1) with

the following initial condition:

cnð0Þ ¼ m1

L

ffiffiffiffiffiffiDk

Nk

rsechðJÞexpðiQÞ; (39)

with

J ¼ m

Lðn � n0Þ; (40)

Q ¼ k þ Dk

2

m2

L2

� �ðn � n0Þ: (41)

In this expression it is assumed that the incident soliton

at t = 0 is centered around site n0 = 1500. The values of

other parameters are set as follows: C = N = 1, L = 1,

m = 0.05, and e = 0.5. Since both C and N are chosen to

be positive, as was discussed above, k must obey

L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101100

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

k

σ

Fig. 2. The dependency of s on k. For the values of the chosen

parameters see the text.

0.4 0.6 0.8 1 1.2 1.40

0.01

0.02

0.03

0.04

0.05

0.06

k

μ A

T(s)

Fig. 3. The dependency of mAðsÞT on k. The squares show the numerical

results for the arithmetic mean of the amplitude maximum and

minimum in the first oscillation of the soliton parameters. The dashed

line represents the theoretical prediction.

|k| < p/2. In Fig. 2 the dependency of s on k is presented

for the given set of parameters.

For the numerical simulation shown in Fig. 1 the

incident pulse has k = 0.8. The corresponding value for

the effective defect strength is s = 0.122, and the

amplitude of the incident soliton is 0.059. It follows

from Eqs. (32) and (33) that |AR| = 0.329 and |AT| = 0.944.

Then, Eq. (28) implies that the reflected pulse is the

dispersive wave packet, while the transmitted pulse

relaxes to the fundamental soliton in the final state. The

amplitude of the generated soliton is:

jcTðt ! 1Þjmax ¼ mAðsÞT ¼ 0:0524; (42)

and, according to Eq. (38), LT = 1.125.

The soliton formation process exhibits a damped

oscillatory behavior of the pulse amplitude and width

around the analytically predicted values. For instance,

the arithmetic mean of the amplitude maximum and

minimum values in the first oscillation is 0.0525. That is

in excellent agreement with Eq. (42). The amplitude of

the reflected pulse decays monotonously with time.

Similarly, for the given width of the incident soliton

L = 1, the excellent agreement between the theory and

numerical simulations is found for 0.4 < |k| < 1.4. For

example, the corresponding analytical predictions and

numerical results for the transmitted soliton amplitude

are given in Fig. 3. For |k| < 0.4 the incident soliton is

too narrow. Indeed, Eq. (24) shows that the incident

wave packet is localized on the scale of order 2L/

m � 40. In turn, for example for k = 0.3, the carrier

wavelength is 2p/k � 21. Therefore, the soliton

envelope is not really slowly varying. In this case,

wider solitons, e.g. with L = 2, must be considered.

Furthermore, the incident soliton is too narrow for

1.4 < |k| < p/2 as well. Although, in this case, |k| < p/2

still holds for the carrier wave number, sufficiently large

portion of the pulse Fourier spectrum does not fall

inside the limits of this range. That causes the

breakdown of the soliton. The increase of pulse width

leads to the shrinkage of its Fourier spectrum, and

therefore, suppresses such instabilities. That is in

agreement with the numerical simulations.

4. Discussion and conclusions

In certain cases the inhomogeneous medium can be

treated as a collection of independent scatterers. That is

often referred to as the independent scattering approach

[34]. For instance, assuming that 0 < s < 3 holds,

consider the soliton tunneling through a set of regular or

irregular set of point defects with the equal strengths e.Moreover, suppose that the soliton width is smaller

compared to the average distance lav between the

neighboring defects. The presented results can be

applied to this problem as well. Indeed, Eq. (38) gives

the soliton width increase after each scattering event.

Then, it is clear that after passing through p defects the

soliton size becomes:

Lð pÞT ¼ L

ffiffiffiffiffiffiffiffiffiffiffi1 þ sp

2 �ffiffiffiffiffiffiffiffiffiffiffi1 þ sp

� � p

:

Therefore, while propagating, the soliton width

increases exponentially. The independent scattering

approach becomes invalid when Lð pÞT lav. Let us as-

sume that this is the case at p = p0. For the system sizes

less than p0lav the transmitted soliton amplitude

decreases exponentially with distance as (see Eq. (37)):

cð pÞT ¼ m

1

L

ffiffiffiffiffiffiDk

Nk

r2ffiffiffiffiffiffiffiffiffiffiffi

1 þ sp � 1

� � p

;

L. Tkeshelashvili / Photonics and Nanostructures – Fundamentals and Applications 11 (2013) 95–101 101

where p < p0. For the larger system sizes the indepen-

dent scattering approach is not valid anymore.

In conclusion, in the weakly nonlinear limit, the

nonintegrable DNLS equation reduces to the integrable

NLS model. This allows to derive the explicit expressions

for the reflected and transmitted pulses from the incident

one. The analytical results suggest that local inhomo-

geneities can be used for the management of the soliton

amplitude and width. In the range of validity, the

theoretical predictions are in excellent agreement with

the numerical simulations. The discussed effects allow

the effective control and manipulation of slow light

pulses, and so, are potentially useful for applications in

the field of all-optical communications.

Acknowledgements

I am grateful to R. Khomeriki for useful discussions.

This work is supported by Science and Technology

Center in Ukraine (Grant No. 5053).

References

[1] F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M.

Segev, Y. Silberberg, Discrete solitons in optics, Physics Reports

463 (2008) 1–126.

[2] S. Longhi, Quantum-optical analogies using photonic structures,

Laser and Photonics Reviews 3 (2009) 243–261.

[3] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals:

Molding the Flow of Light, second ed., Princeton Univ. Press,

Princeton, 2008.

[4] K. Busch, G. von Freymann, S. Linden, S.F. Mingaleev, L.

Tkeshelashvili, M. Wegener, Periodic nanostructures for pho-

tonics, Physics Reports 444 (2007) 101–202.

[5] M. Notomi, E. Kuramochi, T. Tanabe, Large-scale arrays of

ultrahigh-Q coupled nanocavities, Nature Photonics 2 (2008)

741–747.

[6] T.F. Krauss, Why do we need slow light? Nature Photonics 2

(2008) 448–450.

[7] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to

Photonic Crystals, Academic Press, San Diego, 2003.

[8] G.P. Agrawal, Nonlinear Fiber Optics, third ed., Academic Press,

San Diego, 2001.

[9] M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Anal-

ysis and Solitons, Cambridge Univ. Press, Cambridge, 2011.

[10] Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly

integrable systems, Reviews of Modern Physics61 (1989) 763–915.

[11] D. Hennig, G.P. Tsironis, Wave transmission in nonlinear lat-

tices, Physics Reports 307 (1999) 333–432.

[12] D.N. Christodoulides, R.J. Joseph, Discrete self-focusing in nonlin-

ear arrays of coupled waveguides, Optics Letters 13 (1988) 794–796.

[13] D.N. Christodoulides, N.K. Efremidis, Discrete temporal soli-

tons along a chain of nonlinear coupled microcavities embedded

in photonic crystals, Optics Letters 27 (2002) 568–570.

[14] Y. Liu, G. Bartal, D.A. Genov, X. Zhang, Subwavelength

discrete solitons in nonlinear metamaterials, Physical Review

Letters 99 (2007) 153901.

[15] F. Ye, D. Mihalache, B. Hu, N.C. Panoiu, Subwavelength

plasmonic lattice solitons in arrays of metallic nanowires, Phys-

ical Review Letters 104 (2010) 106802.

[16] A. Trombettoni, A. Smerzi, Discrete solitons and breathers with

dilute Bose–Einstein condensates, Physical Review Letters 86

(2001) 2353–2356.

[17] A. Smerzi, A. Trombettoni, Nonlinear tight-binding approxima-

tion for Bose–Einstein condensates in a lattice, Physical Review

A 68 (2003) 023613.

[18] O. Morsch, M. Oberthaler, Dynamics of Bose–Einstein con-

densates in optical lattices, Reviews of Modern Physics 78

(2006) 179–215.

[19] A.B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer,

S. Trillo, S. Wabnitz, Discrete self-trapping, soliton interactions,

and beam steering in nonlinear waveguide arrays, Physical

Review E 53 (1996) 1172–1189.

[20] P.G. Kevrekidis, The Discrete Nonlinear Schrodinger Equation:

Mathematical Analysis, Numerical Computations, and Physical

Perspectives, Springer, Berlin, 2009.

[21] W. Krolikowski, Y.S. Kivshar, Soliton-based optical switching in

waveguide arrays, Journal of the Optical Society of America B

13 (1996) 876–887.

[22] U. Peschel, R. Morandotti, J.S. Aitchison, H.S. Eisenberg, Y.

Silberberg, Nonlinearly induced escape from a defect state in

waveguide arrays, Applied Physics Letters 75 (1999) 1348–1350.

[23] R. Morandotti, H.S. Eisenberg, D. Mandelik, Y. Silberberg, D.

Modotto, M. Sorel, C.R. Stanley, J.S. Aitchison, Interactions of

discrete solitons with structural defects, Optics Letters 28 (2003)

834–836.

[24] M.T. Primatarowa, K.T. Stoychev, R.S. Kamburova, Interaction

of solitons with bond defects in Discrete Nonlinear Schrodinger

chains, Physical Review E 77 (2008) 066604.

[25] A. Fratalocchi, G. Assanto, Governing soliton splitting in one-

dimensional lattices, Physical Review E 73 (2006) 046603.

[26] A.A. Sukhorukov, Y.S. Kivshar, Soliton control and Bloch-wave

filtering in periodic photonic lattices, Optics Letters 30 (2005)

1849–1851.

[27] L. Morales-Molina, R.A. Vicencio, Trapping of discrete solitons

by defects in nonlinear waveguide arrays, Optics Letters 31

(2006) 966–968.

[28] F. Palmero, R. Carretero-Gonzalez, J. Cuevas, P.G. Kevrekidis,

W. Krolikowski, Solitons in one-dimensional nonlinear Schro-

dinger lattices with a local inhomogeneity, Physical Review E 77

(2008) 036614.

[29] Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, S. Bar-

Ad, Nonlinear scattering and trapping by local photonic poten-

tials, Physical Review Letters 99 (2007) 133901.

[30] T. Taniuti, Reductive perturbation method and far fields of wave

equations, Progress of Theoretical Physics Supplement 55

(1974) 1–35.

[31] T. Iizuka, M. Wadati, Scattering of envelope solitons by a mass

impurity in nonlinear lattices, Journal of the Physical Society of

Japan 61 (1992) 4344–4349.

[32] H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D.

Michaelis, A. Brauer, Tailoring guided modes in waveguide

arrays, Optics Express 11 (2003) 3404–3411.

[33] J. Satsuma, N. Yajima, Initial value problems of one-dimensional

self-modulation of nonlinear waves in dispersive media, Prog-

ress of Theoretical Physics Supplement 55 (1974) 284–306.

[34] S.A. Gredeskul, Y.S. Kivshar, Propagation and scattering of

nonlinear waves in disordered systems, Physics Reports 216

(1992) 1–61.